Exponents Chapter Two

Exponents - Chapter Two

• Exponents are a mathematical concept representing repeated multiplication
• Rational exponents are exponents in the form m/n, where m and n are integers

Rational Exponents

• Rational exponents represent roots of numbers
• Example: a^m/n = (ⁿ√a)^m
• Special case: ¹/_n√a = a^1/n

Example 1

• Calculating exponents without a calculator
• Example: 27^2/3 = (3³)^2/3 = 3² = 9

Example 2

• Simplifying expressions with rational exponents
• Example: (81x^-3y⁴)/(16x³y⁴) = (81/16) x^-6y⁰ = (27/4) x^-6

Example 3

• Simplifying expressions involving roots
• Example √x³/√x²

Note

• Coefficients in expressions with roots need to have the root applied to them
• Exponents need to be divided by the order of the root
• Example: √16x⁸= 4x⁴

Exercise 1

• List of practice problems

Simplifying Exponential Expressions

• Methods for simplifying expressions using prime numbers and exponent laws

Example 1 (from Ex 2)

• Simplifying expressions with exponential forms.
• Example: 4^*2x^-2/8x^-1 and 5x⁹x+1 /3x⁺²15x^-1

Example 2 (from Ex 2)

• Simplifying and solving expressions involving roots
• Examples: 2√2 √x²/√16x⁴ and √8 x^-2/√9 x⁴ √62x-9/2√6.

Exercise 2

• List of additional practice problems

Addition and Subtraction

• Simplifying expressions involving addition and subtraction of exponents using factorization

Example 3

• Simplifying expressions using factorization
• Examples: 3x⁺¹-3x⁺²/3x^-3-3^-1 and 3^2x+3x/3^x+1

Other Types of Factorization

• Factoring expressions using differences of squares
• Factoring quadratic trinomials
• Examples: 3^2x-9 and 5^2x+5^x+1-6/ 5^x+6

Example 4

• Simplifying expressions; use of alternative methods
• Examples: 4/2⁺¹ and factorizing with rational exponents;

Example 5

• Simplifying expressions with surds;
• Examples: simplifying √48 and simplifying √54

Example 6

• Simplifying expressions with surds;
• Examples: √2+√8 and (√12+√27)² and (√2-√3)²

Rationalizing the Denominator

• Methods for writing fractions with surds in a form WITHOUT SURDS IN THE DENOMINATOR

Example 7

• Rationalise denominators. Example: 5/√2 and 4/(3-√5)

Exercise 5

• List of practice problems for consolidation

Exponential Equations

• Equations in which the variable is an exponent
• Techniques to solving such equations

The Basic Type

• Making all the bases the same, applying exponent laws and equating exponents

Example 7

• Solving for x: Example 3^-2x = 24 and 9^x-1 = 27.3^x+2 and √4=8x¹/√8x

Using Factorisation

• Identifying common factor type equations
• Examples: 2x+2-2^x = 12 and 3(9^-3^2x-1 = 24

The Quadratic Trinomial Type

• Identifying Quadratic trinomial type equations. Example: 9^x-(4)(3^x)+27=0 and 4^x-3(2^x)+4=0

Exercise 3

• List of practice problems

Equations with Rational Exponents

• Solving equations with rational exponents using reciprocal techniques for evaluating

Example 1.

• Solving for x: x^5/3 = 8

Special Cases

• Cases to be aware of when dealing with even powers or negative values inside roots of expressions. Example x²/n = a where m and/or n are even.

Example 2

• Solve for x ( a) x^2,3=9, (b) x^5/2=32, (c) x^1/3 = -2

Example 3

• solving for x; x^1/3 = 3 and x^5/3 = 32

Example 4

• Solving for x; x-3x²+1 = 0 and x²-3x^-1-2 =0

Exercise 4

• List of practice problems

Test your understanding of exponents and rational exponents in this quiz. Explore concepts such as repeated multiplication, roots of numbers, and how to simplify expressions. Engage with examples that demonstrate both calculation and simplification skills.